Risk Aversion (as opposed to Loss Aversion) is the extent of the preference for certainty in quantifiable, positive outcomes. When presented with a choice between guaranteed and variable outcomes. It is generally assumed, and easily verified, that if the expected outcome is the same, investors prefer a guaranteed outcome to an uncertain one. In fact, it is easy to demonstrate, based on investment choices, that almost everyone will accept a “slightly” lower guaranteed payoff if that avoids any possibility of a worse one. Most explanations for the origin of this behavior are narrative in nature and involve a story like this. When humans first came into being, the threats they faced were often existential in nature. In the absence of hospitals, almost any infection would be fatal. Almost any confrontation with a predator would be fatal. Almost any fire, venomous snake, or hole in the ground would be fatal. As a result, people who over-reacted to risk were more likely to survive long enough to reproduce. After a few thousand generations, the overwhelming majority of humans will be found to be risk-averse, and here we are.
This is a great story, but it can be argued that the original source is much deeper. Andrew Lo and Thomas Brennan offer a more rigorous explanation in a paper from 2011 (Brennan & Lo 2011). Their argument is most easily explained by an example. Consider the first mammal that ever lived. Assume that this critter has 2 options to select from. We will call them Options A and B. The critter can’t really think through the options, he just likes one over the other for whatever reason. Under Option A the critter has exactly 3 offspring, and under Option B there is a 50/50 chance at having either 2 or 4 offspring. Thus, the expected number of immediate descendants is 3 in either case. Let us also assume that this preference can be inherited, meaning that all of the following generations will make the same choice and the population will grow in a perfectly consistent way.
In the absence of variability each critter has 3 offspring, 9 grandchildren, 27 great-grandchildren and so on. This may not sound like much, but consider the fact that rabbits and many other small mammals can give birth twice a year. This means that we get to 20 generations in 10 years. Also, note that 3 to the 20th power is roughly 3.5 Billion. This means that a critter that always takes the sure option will have 3.5 Billion descendants on earth after 10 years.
Now consider the critter that selects Option B. Without knowing it, this critter is accepting risk. This setting under either option is an example of Exponential growth. Under Option A the growth rate is obviously 3, but more formally the growth rate is really Exp(E(ln(x))). Where ln(x) refers to the natural logarithm of x; and E(ln(x)) refers to the expected value of the log of x, and Exp(E(log(x))) is e raised to the power of the expected value of the log of x. (You probably have a headache by now, but bear with me for another 30 seconds.) If x is always 3 we have a growth rate of the natural log of 3 which is 1.06530. Since it’s always this value, this is also the expected value. Finally e1.0653= 3, and this is the rate of growth.
On the other hand, if x has a 50% chance of being 2 and a 50% chance of being 4, the expected value of x, or E(x) is obviously 3. However, the expected value of the ln(x) is ½ * ln(2) + ½ * ln(4). This is strictly less than the ln(3) because the logarithm is a concave function (it becomes flatter as x gets larger.)
Under Option A we have Exp(ln(3)) = 3. Under Option B we get Exp(1/2 * ln(2) + ½ * ln(4)) which is Exp(1/2 * 0.67 + ½ * 1.38) = Exp(1.03972) = 2.82. Since 2.8 is less than 3 we know that the rate of growth is strictly lower. In fact, after 20 generations, this produces a population that is about 215 Million critters smaller. Now, 215 million is not a big number when compared to 3.5 Billion, but remember that this has played out over only 10 years. Mammals came into being around 200 Million years ago. Thus, we had millions upon millions of generations of mammals before humans came along. This means that the population of risk averse mammals overwhelmed any population of mammals that were not risk-averse. Eventually, humans inherited this trait
But we can go a step deeper. I can increase the number of offspring in the risky case to make the outcomes match. We can think of this is a “risk-premium”, meaning an increase in the expected value of offspring in the risky case to compensate for the risk. It turns out that if we use 2.12 and 4.24 instead of 2, and 4, the growth rates match. Lo and behold, this turns out to be exactly 6%. Brennan and Lo point out that the risk premium to get investors to choose a stock portfolio over a risk free portfolio is a little above 6%. (What a funny coincidence!)
In other words, if animals don’t get a substantial “risk premium” in an evolutionary sense, they will be outnumbered by those that always take the sure thing. Millions of years later a potential investor has to see a similar risk premium to entice us to accept the variability inherent in equity markets. What’s the point here? Investors are risk averse, and we won’t accept the risk of equities unless the expected return includes a risk premium. This comes as a surprise to absolutely no one, but it is nice to have a logical explanation for what we see as opposed to simply saying something like, “well that’s just how it is.”
